Let’s say you want to mix a 15% solution of EtOH in water with a 60% solution of EtOH in water to obtain a 20% solution and you need to determine the appropriate volume of each. I know of four different ways to solve for each volume, but by far the easiest is a trick I was taught in a pharmacology course. First you draw a tic-tac-toe grid. Then you write the concentration of the higher strength solution in the upper lefthand corner of the grid. Then you write the concentration of the lower strength solution in the lower lefthand corner of the grid. Then you write the desired concentration in the center square. Now you subtract the desired concentration from the higher strength and write this number in the lower righthand corner, and you subtract the lower concentration from the desired concentration and write this number in the upper righthand corner. So when you’re done, it looks like this.
The numbers in the righthand boxes correspond to the number of parts of each solution—5 parts 60% solution and 40 parts 15% solution will yield a 20% solution. And this trick works for any numbers you plug in, without fail. So how does it work? I’m reasonably adept at algebra, but for the life of me I just can’t visualize in my head how this works. What’s the secret?
The “secret” is just the algebra. Call the concentration of the higher-concentration solution H, and the concentration of the lower one L. Call the desired concentration M.
Then what you’re calculating as the answer is M - L parts of the higher concentration solution to H - M parts of the lower concentration solution. (The tic-tac-toe business is just a particular way of expressing this.)
Why is this the right answer? Well, consider that (M - L) parts + (H - M) parts adds up to H - L parts. And out of this, how much is the solute? It’s (M - L) * H parts of solute from the higher-concentration solution + (H - M) * L parts of solute from the lower-concentration solution = M * H - L * H + H * L - M * L parts of solute = M * (H - L) parts of solute.
So you’re left with M * (H - L) parts of solute out of (H - L) parts total. Which means, you have a concentration of M, as desired.
What you want to do is solve the equation 0.6x + 0.15y = 0.2(x + y) where x is the volume of the stuff that has the 60% concentration, and y is the 15%. So we do some algebra, and we get 60x +15y = 20x +20y. So, now isolate the x’s on one side and the y’s on the other. To do this you take 60x − 20x, which gives you 40x (like your tic-tac-toe board), and 20y − 15y which is 5y. That is where your board gets its numbers. Or more thoroughly, you get the equation 40x = 5y, and the most obvious solution to this is* x* = 5 and y = 40, that is you get 40 × 5 = 40 × 5.
Note that there are many solutions to this equation. You can pick whatever you want for x or y, and if you work the numbers you get an appropriate answer. Note that while 40 and 5 are one possibility, 8 and 1 work as well.
Another way of looking at it is that the problem is really just an instance of “Given two values H and L, what kind of weights should be assigned to them to make their weighted average come out to M?”.
Well, with weights of 1 for the higher and 0 for the lower, the weighted average of course comes out to H, while with weights of 0 for the heights and 1 for the lower, the weighted average of course comes out to L. And as you move the weights between those two, the weighted average moves linearly between those outcomes of H and L.
Accordingly, the ratio of the weighted average’s distance from L to its distance from H is always equal to the ratio of the weights assigned to H and to L. Which is precisely the rule “To get a weighted average of M out of H and L, use weights of (H - M) for L and (M - L) for H”.
No - the secret is part algebra and part magic. To figure out the exact ratio of algebra to magic first you draw a tic-tac-toe grid, then you write the fraction of common-sense in the upper-left corner and then you write the amount of head-scratching awe in the lower-left corner…
